Thermodynamics of electrical noise in a class of nonlinear RLC networks

This paper addresses the equilibrium and transient behavior of a class of nonlinear
RLC
circuits driven externally by deterministic inputs and internally by thermal noise from linear resistors. Resistor noise is described by the standard Nyquist-Johnson model [1], [2]. Physical principles from thermodynamics are formulated as theorems concerning the stochastic differential equation and the associated forward Kolmogorov equation describing the network and are proved on a rigorous basis. The forward Kolgomorov equation governing the evolution of the probability density for certain capacitor charges and inductor fluxes is shown to be an infinite dimensional dissipative dynamical system in the sense of Willems [3]. By this route we demonstrate that essentially all the principles of thermodynamics for this class of systems can be derived as mathematical consequences of the Nyquist-Johnson model for thermal noise in linear resistors. A significant clarification of thermodynamic theory results from this formulation, since the mathematical framework and a number of specific conclusions are valid for transient as well as equilibrium behavior.